The exact correspondence between ordinal notations derived from Skolem hull operators, which are classical in ordinal analysis, and descriptions of ordinals in terms of Σ1-elementarity, an approach developed by T.J. Carlson, is analyzed in full detail. The ordinal arithmetical tools needed for this purpose were developed in [G. Wilken, Ordinal arithmetic based on Skolem hulling, Annals of Pure and Applied Logic 145 130–161]. We show that the least ordinal κ such that κ<1∞ 19–77] and described below) is the proof theoretic (...) ordinal of the set-theoretic system , confirming a claim of Carlson. Moreover, we characterize the class of all ordinals κ such that κ<1∞ and provide an ordinal arithmetical analysis of Carlson’s entire structure in the style of [T.J. Carlson, Ordinal arithmetic and Σ1-elementarity, Archive for Mathematical Logic 38 449–460]. (shrink)

How do ordinals measure the strength and computational power of formal theories? This paper is concerned with the connection between ordinal representation systems and theories established in ordinal analyses. It focusses on results which explain the nature of this connection in terms of semantical and computational notions from model theory, set theory, and generalized recursion theory.

We will study patterns which occur when considering how Σ 1 -elementary substructures arise within hierarchies of structures. The order in which such patterns evolve will be seen to be independent of the hierarchy of structures provided the hierarchy satisfies some mild conditions. These patterns form the lowest level of what we call patterns of resemblance . They were originally used by the author to verify a conjecture of W. Reinhardt concerning epistemic theories 449–460; Ann. Pure Appl. Logic, to appear), (...) but their relationship to axioms of infinity and usefulness for ordinal analysis were manifest from the beginning. This paper is the first part of a series which provides an introduction to an extensive program including the ordinal analysis of set theories. Future papers will conclude the introduction and establish, among other things, that notations we will derive from the patterns considered here represent the proof-theoretic ordinal of the theory KPℓ 0 or, equivalently, Π 1 1 −CA 0. (shrink)

We will investigate patterns of resemblance of order 2 over a family of arithmetic structures on the ordinals. In particular, we will show that they determine a computable well ordering under appropriate assumptions.

We study the structure of families of theories in the language of arithmetic extended to allow these families to refer to one another and to themselves. If a theory contains schemata expressing its own truth and expressing a specific Turing index for itself, and contains some other mild axioms, then that theory is untrue. We exhibit some families of true self-referential theories that barely avoid this forbidden pattern.

Reinhardt’s conjecture, a formalization of the statement that a truthful knowing machine can know its own truthfulness and mechanicalness, was proved by Carlson using sophisticated structural results about the ordinals and transfinite induction just beyond the first epsilon number. We prove a weaker version of the conjecture, by elementary methods and transfinite induction up to a smaller ordinal.

We construct a machine that knows its own code, at the price of not knowing its own factivity.

Elementary patterns of resemblance notate ordinals up to the ordinal of Pi^1_1-CA_0. We provide ordinal multiplication and exponentiation algorithms using these notations.

We propose that, for the purpose of studying theoretical properties of the knowledge of an agent with Artificial General Intelligence (that is, the knowledge of an AGI), a pragmatic way to define such an agent’s knowledge (restricted to the language of Epistemic Arithmetic, or EA) is as follows. We declare an AGI to know an EA-statement φ if and only if that AGI would include φ in the resulting enumeration if that AGI were commanded: “Enumerate all the EA-sentences which you (...) know.” This definition is non-circular because an AGI, being capable of practical English communication, is capable of understanding the everyday English word “know” independently of how any philosopher formally defines knowledge; we elaborate further on the non-circularity of this circular-looking definition. This elegantly solves the problem that different AGIs may have different internal knowledge definitions and yet we want to study knowledge of AGIs in general, without having to study different AGIs separately just because they have separate internal knowledge definitions. Finally, we suggest how this definition of AGI knowledge can be used as a bridge which could allow the AGI research community to import certain abstract results about mechanical knowing agents from mathematical logic. (shrink)